Discrete bispectral Darboux transformations from Jacobi operators

By F. Alberto Grünbaum, M. Yakimov. 2000

69 citations recorded by the source corpus.

We construct families of bispectral difference operators of the form a(n)T + b(n) + c(n)T -1 where T is the shift operator. They are obtained as discrete Darboux transformations from appropriate extensions of Jacobi operators. We conjecture that along with operators previously constructed by Grunbaum, Haine, Horozov and Iliev they exhaust all bispectral regular (i.e., a(n) ¬= 0, c(n) ¬= 0, Vn E Z) operators of the form above.

Subfields: Algebra and Number Theory, Geometry and Topology, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/cf2100e01854bc1000f5b982c9105e9cb77dec33

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Algebra and Number Theory

1. Jeffrey S. Geronimo
2. Plamen Iliev
3. Arno B. J. Kuijlaars

In Geometry and Topology

1. Jeffrey S. Geronimo
2. Plamen Iliev

In Statistical and Nonlinear Physics

1. Jeffrey S. Geronimo
2. Plamen Iliev

Citations in the corpus

• Bispectrality of Multivariable Racah–Wilson Polynomials (2007, methodology): “In the one dimensional case, this involution played a crucial role in the construction of orthogonal polynomials satisfying higher-order differential or q-difference equations, by applying the Darboux transformation to the second order difference operator L corresponding to the Jacobi polynomials [1”